My first bike since university was a
Condor Pista, bought about 10 years ago. My first riding experience was taking it back from the shop through central London. It was fine because I did not go fast and did not have clip-ins for a while.
I see other people with flip-flop wheels going single speed but never transitioning to fixed from free-wheel once they start a single speed set-up.
My advice is just take it easy but go straight to fixed. Do have front and back brakes (for emergency use) and put some flat pedals on until you get used to it. I switch between geared and fixed and often have momentary lapses going round a corner but I think you'll find that while the momentum you have is so powerful that your legs have no choice, your body is so heavy that you won't get evicted from your saddle unless you are really pegging it. So these lapses don't unnerve me.
Focus on riding at slower speeds to start, with accelerations and decelerations without using brakes and you will soon start feeling the bike takes you in smoother more defensive lines through traffic, around other cyclists and at junctions. It is good fun.